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An arithmetic analogue of the theorem of Bézout. (Un analogue arithmétique du théorème de Bézout.) (French) Zbl 0756.14012
For a closed subvariety $$X$$ in a projective space defined over the ring of rational integers
G. Faltings defined the height $$h(X)$$ and showed $$h(X)\geq 0$$. $$h(X)$$ coincides with the logarithmic height of a generic point of $$X$$.
First the authors show that the height of $$X$$ is bounded below by the geometric degree of $$X$$. Next they show the height of the proper intersection of two subvarieties is bounded above by the height and the degree of each variety. This formula is an arithmetic analogue of the theorem of Bézout.
Reviewer: H.Maeda (Tokyo)

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 11G50 Heights